Topology of \(\mathbb {R}P^3\)  can be understood in many ways. The simplest description is to think of it is as a compactification of \(\mathbb {R}^3\)  by adding a projective plane at infinity. Since,
the complement of an open 3-ball in \(\mathbb {R}^3\)  is homeomorphic to \(S^2\times [0,1)\) , once we add an \(\mathbb {R}P^2\)  at infinity the complement of a 3-ball in \(\mathbb {R}P^3\)  is homeomorphic to the space obtained by compactifying \(S^2\times [0,1)\)  by adding an \(\mathbb {R}P^2\) , which is nothing but the mapping cylinder ([1]} ) of the two sheeted covering \(S^2\rightarrow \mathbb {R}P^2\) . Thus \(\mathbb {R}P^3\)  can also be thought of as, joining a 3-ball to this mapping cylinder along their boundary sphere. We will use this decomposition frequently.
